Quasicrystalline structures and uses thereof

ABSTRACT

This invention relates generally to the field of quasicrystalline strictures, In preferred embodiments, the stopgap structure is more spherically symmetric than periodic structures facilitating the formation of stopgaps in nearly all directions because of higher rotational symmetries. More particularly, the invention relates to the use of quasicrystalline structures for optical, mechanical, electrical and magnetic purposes. In some embodiments, the invention relates to manipulating, controlling, modulating and directing waves including electromagnetic, sound, spin, and surface waves, for pre-selected range of wavelengths propagating in multiple directions.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of, and claims priority to,co-pending U.S. patent application Ser. No. 15/410,396 filed Jan. 19,2017, which claims priority to Ser. No. 14/059,003 filed Oct. 21, 2013,now U.S. Pat. No. 9,567,420 issued on Feb. 14, 2017, which claimspriority to U.S. patent application Ser. No. 13/906,792 filed May 31,2013, now U.S. Pat. No. 8,599,472 issued on Dec. 3, 2013, which claimspriority to Ser. No. 13/553,149 filed Jul. 19, 2012, now U.S. Pat. No.8,508,838 issued on Aug. 13, 2013, which claims priority to U.S. patentapplication Ser. No. 13/271,969 filed Oct. 12, 2011, now U.S. Pat. No.8,243,362 issued on Aug. 14, 2012, which claims priority to U.S. patentapplication Ser. No. 11/988,480 filed Apr. 6, 2009, now U.S. Pat. No.8,064,127 issued on Nov. 21, 2011, which is a national state entry ofPCT/US2006/026430 filed Jul. 7, 2006, now expired, which claims priorityto Provisional Application Ser. Nos. 60/697,829 filed on Jul. 8, 2005and 60/697,872 filed on Jul. 8, 2005 now expired, the contents of whichare incorporated herein in their entirety.

GOVERNMENT RIGHTS

This invention was made with government support under Grant No.DMR0243001 and Grant DMR0213706 awarded by the National ScienceFoundation, and under Grant No. DE-FG02-91ER40671 awarded by theDepartment of Energy. The government has certain rights in theinvention.

FIELD OF THE INVENTION

This invention relates generally to the field of quasicrystallinestructures. In preferred embodiments, the stopgap structure is morespherically symmetric than periodic structures facilitating theformation of stopgaps in nearly all directions because of higherrotational symmetries. More particularly, the invention relates to theuse of quasicrystalline structures for optical, mechanical, electricaland magnetic purposes. In some embodiments, the invention relates tomanipulating, controlling, modulating and directing waves includingelectromagnetic, sound, spin, and surface waves, for a pre-selectedrange of wavelengths propagating in multiple directions.

BACKGROUND OF INVENTION

The speed of electronic devices is achieved by miniaturizing electroniccomponents to a very small micron-size scale so that those electronsneed to travel only very short distances within a very short time.Current technology is approaching its fundamental limits in thesub-micron miniaturization process. Further miniaturization introducesseveral technological problems such as dielectric breakdown, hotcarriers, and short channel effects. Optical interconnections andoptical integrated circuits provide a way out of these limitations tocomputational speed and complexity inherent in conventional electronics.Optical components use photons traveling in waveguides instead ofelectrons to perform the appropriate functions. Optical circuits guide,redirect, trap, and manipulate photons. Thus, there is a need to developmaterials to facilitate these actions.

Present technology relies on physical components whose size is largecompared to the wavelength of the light being transmitted. For example,currently, light is bent by 90 degrees by guiding it along an opticalfiber whose the bend radius is millimeters or greater. By contrast, withphotonic crystals, light propagates down straight channels and can beredirected around corners which are a thousand times smaller (a micronin size), thus enabling the entire circuit to be miniaturized. Thephotonic crystal can also be modified to trap and manipulate thephotons. Hence, there is a need to develop photonic crystal componentsso that photonic circuits can process photons in a way comparable to howelectronic circuits process electrons.

SUMMARY OF THE INVENTION

This invention relates generally to the field of quasicrystallinestructures. In preferred embodiments, the stopgap structure is morespherically symmetric than periodic structures facilitating theformation of stopgaps in nearly all directions because of higherrotational symmetries. More particularly, the invention relates to theuse of quasicrystalline structures for optical, mechanical, electricaland magnetic purposes. In some embodiments, the invention relates tomanipulating, controlling, modulating and directing waves includingelectromagnetic, sound, spin, and surface waves, for a pre-selectedrange of wavelengths propagating in multiple directions.

In some embodiments, the invention relates to a quasicrystallinestructure with quasi-periodic translational symmetry, i.e., anicosahedral quasicrystal, which can be used to filter, stop, redirect,modify and manipulate the propagation of light in user-selectedfrequency ranges, with closer gap centers and more spherically symmetricstopgap than ordinary periodic crystals. It is the intent of theapplicants that the structures can be rescaled to any size or geometrydepending on what frequencies are desired.

In some embodiments, the invention relates to observing the changes ofelectromagnetic transmission through quasicrystalline structures byvarying electromagnetic frequencies and direction. In furtherembodiments, the invention relates to a quasicrystalline material with acomplete bandgap.

In some embodiments, the invention relates to a structure whosedielectric constant can be expressed as sum of periodic waves withwavefronts along the symmetry axes of an icosahedron and where thedielectric constant may change continuously or be constructed from amixture of two or more dielectric materials whose geometrical shapeshave different numbers of extended and small dimensions (e.g., a mixtureof rods, ellipsoids, and planes). In further embodiments, the symmetryaxis has other non-crystallographic symmetries preferably five-fold,seven-fold, or greater than seven-fold, and all symmetries in the planeand either periodic or quasiperiodic in the third direction.

In some embodiments, the invention relates to a structure whosespatially modulated (or varying) dielectric constant (or index ofrefraction) can be expressed as sum of periodic waves oriented alongdirections arranged with icosahedral symmetry and where the dielectricconstant is smoothly varying or the interface between dielectrics issmoothly varying or is approximated by two or more geometrical(topological) shapes (e.g., rods, ellipsoids, and planes).

In some embodiments the invention relates to a material wherein adielectric constant may be continuously varying in a quasicrystallinepattern through the medium or the structure maybe composed of discreteelements (rods, ellipsoids and planes) whose placement approximates asmoothly changing density or in which two or more geometrical shapes areused in order to block both polarizations (TE and TM) of photons (i.e.,not a single element, like rods only or spheres only).

In some embodiments, the invention relates to a material comprising aplurality of refractive indices at different locations in the materialwherein said refractive indices have a plurality of maximum values and aplurality of minimum values and wherein said plurality of maximum andminimum values are located in a pattern that is quasiperiodic. Infurther embodiments, said pattern is quasiperiodic in three-dimensionalspace. In further embodiments, said pattern produces an icosahedralshape. In further embodiments, the arrangement is comprised five-fold,seven-fold or greater-fold symmetry axes. In further embodiments, one ofthe pluralities of refractive indices of a material (one refractiveindex) does not include any radiation that travels through air orvacuum. In further embodiments, one of the pluralities of refractiveindices does not include slight variations at the material'smaterial-gas interface or material-vacuum interface. In furtherembodiments, the material has slight imperfections at the interfacepreferably between substances providing differing electromagneticproperties preferably differing refractive indices.

In some embodiments, the invention relates to a structure comprising afirst material that is separated into a plurality of units and a secondmaterial wherein said first material has a refractive index differentfrom the refractive index of said second material wherein said secondmaterial interconnects said plurality of units of said first materialwherein said plurality of units of said first material is arranged in apattern that is quasicrystalline. In further embodiments, said pluralityof units are silica spheres. In further embodiments, said patternproduces an icosahedral shape. In further embodiment, said secondmaterial is a polyacrylamide hydrogel.

In some embodiment, the invention relates to a structure comprised of afirst plurality of geometrical shapes and a second plurality ofgeometrical shapes wherein said first plurality of geometrical shapesare arranged in a pattern that is quasicrystalline and wherein thesecond plurality of geometrical shapes are arranged to interconnect thefirst plurality of geometrical shapes. In further embodiments, saidfirst plurality of geometric shapeS are spheres. In further embodiments,said second plurality of geometric shapes are rods. In furtherembodiments, said pattern produces an icosahedral shape. In furtherembodiments, in addition to spheres and rods, one may substitutesurfaces, sheets, stars, rhombohedra and other geometrical shapes.

In some embodiments, the invention relates to a method of detectingelectromagnetic radiation comprising: a) providing: i) aquasicrystalline structure and ii) electromagnetic radiation; b)directing said electromagnetic radiation through said quasicrystallinestructure; and c) detecting said electromagnet radiation.

In some embodiments, the invention relates to a method of creatingdirectionally oscillating electromagnetic radiation comprising: a)providing: i) a quasicrystalline structure ii) a voltage iii)electromagnetic radiation b) applying said voltage across saidquasicrystalline structure under conditions such that saidquasicrystalline structure oscillates electromagnetically; and c)directing said electromagnetic radiation through said quasicrystallinestructure.

In some embodiments, the invention relates to a waveguide comprising aphotonic quasicrystal that can guide incoming light at any angle. Infurther embodiments, the invention relates to a waveguide comprising aphotonic quasicrystal formed on the top of a light emitting diode. Infurther embodiments, the photonic quasicrystal may guide light enteringthe photonic quasicrystal from any direction.

In some embodiments, the invention relates to a composition comprising alattice of rods configured such that they interconnect a plurality oflattice points arranged in quasi-periodic pattern.

In some embodiments, the invention relates to a method of detectingphotonic band-gaps comprising: i) providing a) a lattice comprising aplurality of optical rods wherein said plurality of rods is configuredsuch that said rods have intersections that form a plurality of latticepoints in a quasi crystalline pattern; b) incident electromagneticradiation; and c) an instrument capable of detecting exitingelectromagnetic radiation; ii) directing said incident electromagneticradiation through said lattice; and iii) detecting said exitingelectromagnetic radiation from said lattice with said instrument.

In some embodiments, the invention relates to a composition comprising alattice of intersecting transparent rods configured such that they forma plurality of lattice points arranged in a quasi-crystalline pattern,

In some embodiments, the invention relates to a material comprising afirst pattern that is quasicrystalline and defects that are arranged ina second pattern designed to direct and control the radiation.

In some embodiments, the invention relates to a complete bandgap thathas the symmetry of the photonic quasicrystal, which can be chosen to benearly circularly symmetric in two dimensions or nearly sphericallysymmetric in three dimensions.

In some embodiments, the invention relates to a material comprising afirst pattern that is quasicrystalline and defects that are a secondpattern.

In some embodiments, the invention relates to a composition of matterwhich comprises a quasicrystalline arrangement of materials, saidmaterials having a quasiperiodically ordered pattern with apreselectable rotational symmetry having a characteristic photonicbandgap structure forbidden in a crystalline material. In furtherembodiments, the quasicrystalline structure comprises at least twodielectric materials. In further embodiments, the quasicrystallinestructure includes higher point group symmetry than a crystallinecounterpart. In further embodiments, a complete bandgap is substantiallyspherically symmetric and/or substantially circularly symmetric.

In some embodiments, the invention relates to a lamp having a lightsource comprising a photonic quasicrystal designed to produce radiationsubstantially only within the visible spectrum or any finitepre-selected frequency range of the electromagnetic spectrum.

In some embodiments, the invention relates to a photonic quasicrystalreflector. In further embodiments, said quasicrystal comprises a two- orthree-dimensional quasicrystalline array of dielectric elements.

In some embodiments, the invention relates to a compact optical circuitcomprising a photonic quasicrystal reflector.

In some embodiments, the invention relates to an optical filtercomprising a dielectric layer formed within a resonant optical cavity,with the dielectric layer having formed therein a sub-wavelengthquasicrystalline structure to define, at least in part, a wavelength fortransmission of light through the resonant optical cavity, wherein saidquasicrystalline structure is a photonic quasicrystal.

In some embodiments, the invention relates to a broadband antenna systemcomprising multiple photonic bandgap quasicrystals for achieving higherpower efficiency over a selected range of frequencies, wherein each ofsaid quasicrystals is designed to cover a specific range of frequencies.In further embodiments, said multiple quasicrystals are attachedtogether to form a photonic bandgap substrate whose bandwidth varies asa function of location on the substrate.

In some embodiments, the invention relates to a photonic bandgap mirrorcomprising a quasicrystalline lattice structure.

In some embodiments, the invention relates to a stealth materialcomprising photonic quasicrystals designed to trap electromagneticradiation at programmable ranges of frequencies. In further embodiments,said radiation is trapped nearly uniformly at all angles.

In some embodiments, the invention relates to a dielectric resonatorcomprising a resonant defect structure placed in a quasicrystal latticeof dielectric elements

In some embodiments, the invention relates to a quasicrystal,quasicrystals or quasicrystal lattice wherein said quasicrystalcomprises structural defects introduced by removing or positioningparticles or elements in the quasicrystal in a manner to thereby controlthe optical and physical properties of said quasicrystal.

In some embodiments, the invention relates to a composition of mattercomprising a quasicrystal wherein said quasicrystal comprises structuraldefects introduced by removing or positioning particles or elements inthe quasicrystal in a manner to thereby control the optical and physicalproperties of said quasicrystal.

In some embodiments, the invention relate to a method of making aphotonic bandgap material which comprises arranging dielectric elementsor materials in a quasicrystalline order, said photonic bond gapmaterial comprising a quasiperiodically ordered pattern with apreselectable rotational symmetry and a characteristic photonic bandgapstructure forbidden in a crystalline material. In further embodiments,said quasicrystalline order comprises structural defects introduced byremoving or positioning particles or elements in the quasicrystal in amanner to thereby control the optical and physical properties of saidquasicrystal.

In some embodiments the invention relates to the use of a quasicrystalto produce a photonic bandgap material.

In some embodiments, the invention relates to an icosahedral photonic zoquasicrystal.

The present invention relates to photonic quasicrystals, and preferablythose that can be used to make photonic bandgap materials. Thesephotonic quasicrystals have many uses, including the same uses asphotonic crystals.

It is therefore an object of certain embodiments of the invention toprovide an improved system and method for fabricating photonic stopgapstructures by using a quasicrystalline arrangement of elements.

It is also an object of certain embodiments of the invention to providean improved article of manufacture of a three dimensional photonicstructures by using a quasicrystalline arrangement of elements.

It is a further object of certain embodiments of the invention toprovide an improved system and method for constructing materials havingphotonic bandgaps forbidden in crystalline materials.

It is yet another object of certain embodiments of the invention toprovide an improved system and method for constructing rotationallysymmetric structures having optical, structural and mechanicalproperties unachievable by crystalline materials.

It is an additional object of certain embodiments of the invention toprovide an improved quasicrystalline structure having programmableoptical, structural, mechanical and chemical properties.

It is also another object of certain embodiments of the invention toprovide an improved system and method for constructing a photonicstopgap structure with programmable Brillouin zones for selectedtechnological applications.

It is also a further object of certain embodiments of the invention toprovide an improved system and method for constructing a photonicstopgap material with substantially more spherical Brillouin zones.

It is also a further object of certain embodiments of the invention toprovide an improved system and method for constructing a photonicstopgap material with substantially more circular Brillouin zones in twodimensions.

It is yet an additional object of certain embodiments of the inventionto provide an improved system and method for constructing aquasicrystalline material having long range orientational order withouttransitional periodicity and constructed to operate in a predeterminedmanner responsive to electromagnetic radiation.

It is also a further object of certain embodiments of the invention toprovide an improved system and method for constructing photonic stopgapstructures which can be switched from one structural state to anotherstate by repositioning particles to thereby modify physical and/orchemical properties of the arrangement.

It is also a further object of certain embodiments of the invention toprovide an improved system and method for constructing photonic stopgapstructures in which structural defects are introduced by removing orpositioning particles in the quasicrystal to control the optical andphysical properties.

It is still another object of certain embodiments of the invention toprovide an improved system and method for constructing photonic stopgapstructures by use of optical devices to dynamically modify properties inaccordance with time sensitive requirements.

It is yet another object of certain embodiments of the invention toprovide an improved system and method for organizing and connectingdifferent linear and nonlinear components in a quasicrystallinestructure for large scale integration of optical circuits in two andthree dimensions.

It is another object of certain embodiments of the invention to providean improved system and method for constructing photonic stopgapheterostructures using quasicrystalline arrangements of elements to formengineered features which enable creation of narrow band waveguides andfrequency selective filters of electromagnetic radiation.

It is yet another object of certain embodiments of the invention toprovide an improved system and method for organizing disparatecomponents to position selectable components in a quasicrystallinestructure for establishing chemical and physical properties for adesired technological application

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A-D depicts experimental photonic structures and their Brillouinzones. FIG. 1a shows stereolithographically produced icosahedralquasicrystal with 1 cm length rods. FIG. 1b shows a diamond structurewith 1 cm rods. FIG. 1c is a drawing of a triacontahedron, one ofseveral possible effective Brillouin zones of icosahedral symmetry. FIG.1d shows the Brillouin zone for the FCC/diamond structure.

FIG. 2A-B illustrates the measured transmission for an icosahedralquasicrystal. FIG. 2a plots T(f,θ), transmission as a function offrequency and angle, for a rotation about a two fold rotation axis ofthe quasicrystal (corresponding to the dotted line in FIG. 1c ) usingtwo overlapping frequency bands. The dashed line is a 1/cos(θ) curvecharacteristic of Bragg scattering from a Brillouin zone face. FIG. 2bplots T(f,θ) for a rotation about a 5-fold rotation axis correspondingto the dashed line in FIG. 1 c. FIG. 2c shows a schematic of themicrowave horn and lens arrangement used for these measurements.

FIG. 3A-B illustrates a comparison of calculated bands and measuredtransmission for a diamond structure. FIG. 3a depicts the calculateddispersion relation, f, on the boundary of the first Brillouin zone vs.θ, for the diamond structure along the dotted curve in FIG. 1 d. FIG. 3bplots T(f,θ) for the sample rotation along the same curve. There isagreement in the photonic gap center frequencies.

FIG. 4A-E provides the imaging of Brillouin zone for diamond andicosahedral quasicrystal structures. FIG. 4a depicts the Brillouin zonefor the diamond structure along a 4-fold direction as seen in contourplot of calculated frequency deviation (δf=f−(c/n)|k|/(2π)) vs. k. FIGS.4b-e depict the Brillouin zone in plot of the measured T(r=f), θ=θ)(using the same scale as FIG. 3A-B) for the diamond lattice along the(b) 4-fold (dashed in FIG. 1d ) and (c) 2-fold (dotted in FIG. 1d )axis; and for the quasicrystal along the (d) five-fold (dashed in FIG.1c ) and two-fold (dotted in FIG. 1c ) axes. The inner decagon in (d)and the solid and dashed lines in (e) correspond to dashed and dottedlines in FIG. 1c . The dash-dotted line is a non-triacontahedral zoneface.

DETAILED DESCRIPTION OF THE INVENTION

This invention relates generally to the field of quasicrystallinestructures. In preferred embodiments, the stopgap structure is morespherically symmetric than periodic structures facilitating theformation of stopgaps in nearly all directions because of higherrotational symmetries. More particularly, the invention relates to theuse of quasicrystalline structures for optical, mechanical, electricaland magnetic purposes. In some embodiments, the invention relates tomanipulating, controlling, modulating and directing waves includingelectromagnetic, sound, spin, and surface waves, for a pre-selectedrange of wavelengths propagating in multiple directions.

Photonic crystals may have stopgaps that direct which frequencies anddirections of light are blocked or allowed to pass through. For example,in a cubic photonic crystal, stopgaps may block light of certainfrequencies from exiting its faces while allowing the same frequency oflight to escape along its edges. A complete photonic bandgap resultswhen stopgaps overlap in all directions and, for a certain range offrequencies, light is completely trapped. A limitation of photoniccrystals is that their periodic translational symmetry constrains thestopgap structure to block only certain directions and frequencies.Quasicrystals have rotational symmetries disallowed for crystals.Quasicrystalline structures can have arbitrary rotational symmetries intwo and three dimensions. Because quasicrystals can have higherrotational symmetries, their stopgap structure can be more sphericallysymmetric than that of any photonic crystal, facilitating the formationof stopgaps in all directions and enabling practical devices to beconstructed from a wider variety of materials. Furthermore, because theyhave quasiperiodic (rather than periodic) translational symmetry, theycan have more diverse and controllable stopgap structures.

A “pattern” means a two- or three-dimensional structure that can beconstructed by either: (a) repeating a finite number of bounded designmotives, structural elements, or building blocks joined along theirboundaries or overlapping one another; or (b) overlaying regularsinusoidal waves of density or other physical property (such asdielectric constant) in which the waves are oriented in differentdirections.

A “location” means a point or extent in space. A pattern is“periodically ordered” or “crystalline” or “crystal ordered” if it canbe constructed either: (a) from a single design motif, structuralelement or building block repeated with equal spacing betweenrepetitions joined together along their boundaries without overlappingone another and the combined pattern only has two-, three-, four- orsix-fold symmetry axes; or (b) overlaying regular sinusoidal waves ofdensity or other physical property (such as dielectric constant) inwhich the waves are oriented along only directions where the combinedwave pattern has only two-, three-, four- or six-fold symmetry axes. Theterms “crystal pattern” or “crystalline pattern” are synonyms forperiodic pattern.

Periodic means repeating at regular (equal) intervals. A “period” is theinterval between repetitions. As used herein, the term “quasiperiodicpattern” often refers to the quantity of some physical characteristic ofmatter, like the density or dielectric constant or spin or otherfeature. A pattern is “quasiperiodic” if it can be decomposed intorepeating elements or waves with more than one frequency in which theratios of frequencies is irrational, or if it can be decomposed in a sumof regularly sinusoidal waves whose crests overlap at more than onefrequency in which the ratios of frequencies is irrational. Rationalmeans the numbers are expressible as a ratio of integers, such as 5/3.Irrational means numbers are not exactly expressible as a ratio ofintegers, such as, but not limited to, a ratio containing the squareroot of two. Irrational numbers have decimal expansions that neitherterminate nor become periodic. It is the intent of the applicants thatthe term “irrational” with relation to quasiperiodic includeapproximants of irrationals as well. For example, a rational approximantis a ratio of integers P/Q whose value is nearly equal to the value ofthe irrational. For example, the golden ratio is (1+√5)/2=1.618033 . . .[and so on as there is no finite decimal]. A rational approximant is21/13=1.61538, since no ratio with denominator less than 13 comes closerto the actual value. One can construct a series of rational approximantsthat come as close as you want to an irrational number (but not quiteequal it). For example, 34/21=1.61905, 55/34=1.61765, and233/144=1.61806.

Quasiperiodic includes a quantity that is not periodic itself but thatcan be expressed as a finite sum of periodic quantities whose periodshave ratios that are irrational. An example is x[n]=n(1+(1/t²))+(1/t){{n/t}}, where t=(1+√[5])/2=golden ratio (an irrational number) andwhere {{ ••}} means truncate the integer part of the quantity inside andkeep only the decimal part. The function x[n] describes a quasiperiodicspacing of points in one dimension in which the first term is periodicwith period (1+(1/t²)) and the second term is periodic with period t.The ratio of the two periods is an irrational number. This examplecorresponds to a “Fibonacci” lattice of distances, which can be seen insome directions in a two dimensional Penrose tiling and also thethree-dimensional icosahedral quasicrystal. In another example,quasiperiodic in D dimensions means a quantity that is quasiperiodicalong any line in D dimensions.

A pattern is “quasicrystalline” or “quasicrystal ordered” if it isquasiperiodic and has at least one five-, seven-, or higher-foldsymmetry axis. A quasicrystal pattern may be constructed either: (a)from a finite number of distinct design motives, structural elements, orbuilding blocks each of which regularly repeats with an independentaverage spacing and where the ratios of the spacings is irrational andthe pattern includes five-, seven- or any higher-fold symmetry axes; or(b) overlapping regular sinusoidal waves of density or other physicalproperty (such as dielectric constant) in which the waves are orientedalong directions where the combined pattern includes five-, seven- orany higher-fold symmetry axes.

An infinite periodic pattern has the property that two overlappingcopies of the pattern can be shifted with respect to one another by afinite distance and overlap once again. An infinite periodic patternalso has the property that two overlapping patterns can be rotated by afinite angle along a discrete set of axes and overlap once again. Thediscrete set of angles which result in this overlap defines therotational symmetry of the periodic pattern, which, according to thewell-known rigorous mathematical theorems of crystallography, onlyallows two-, three-, four- and six-fold rotational symmetry in the planeand two-, three-, four- and six-fold symmetry axes in three dimensions.A finite piece of an infinite periodic pattern is also called a periodicpattern, provided it contains a multiplicity of the repeating motives,structural elements, or building blocks.

An infinite quasiperiodic pattern made of discrete elements has theproperty that two overlapping copies of the pattern cannot be shiftedwith respect to one another by a finite distance and precisely overlapthroughout. An infinite quasiperiodic pattern made of discrete elementsalso has the property that a sequence of finite shifts can be found sothat the two copies overlap out to longer and longer distances. Aninfinite quasicrystalline pattern also has the property that it can berotated by finite set of angles along a discrete set of axes so that theorientations of the design motives, structural elements, building blocksor sinusoidal waves composing the rotated pattern are the same as theoriginal pattern. The finite set of angles which define the rotationalsymmetry of the quasicrystalline pattern, which must include at leastone five-, seven-, or higher-fold symmetry axes. A finite piece of aninfinite quasicrystalline pattern is also called a quasicrystallinepattern, provided it contains a multiplicity of the repeating motives,structural elements, or building blocks.

An example of periodic order in two-dimensional space is squares thatmake up a grid. Two overlapping square grids can be shifted with respectto one another by the length of one edge, or a multiple thereof, and thetwo grids will overlap again. Also, squares have 4-fold symmetry.Another example of a periodic pattern would be a grid of hexagons thathas 6-fold symmetry. However, one cannot create a two-dimensional gridof solely true pentagons without producing a spatial gap or overlap.Also, one cannot construct a pattern with periodic order with five-,seven- or any higher-fold symmetry axes.

An example of a quasicrystalline pattern is the two-dimensional Penrosepattern composed of acute and obtuse rhombi (diamond shapes)—an acuteone with a small angle of 36 degrees and the obtuse one with a smallangle of 72 degrees—in which the acute and obtuse rhombi repeat withdifferent average frequencies whose ratio is the golden ratio,(1+√{square root over (5)})/2, an irrational number, and in which thepattern has an average ten-fold symmetry. As for all quasicrystallinepatterns, there cannot be more than a single point of “exact” infinitelyexpanding symmetry. This Penrose pattern is quasiperiodic.

For example, the Penrose pattern is decagonal even though it containsnot even a single point of “exact” 10-fold symmetry. If a red Penrosepattern is placed over an identical blue one and the red pattern is thenrotated counterclockwise, nothing interesting happens until a 10-fold(36 degree) rotation is completed and a fair fraction of the vertices ofthe two patterns coincide. The red tiling is then translated by someamount until whole regions, of about 10 to 15 patterns across, coincide.Finally, the red pattern is translated even further until regions of theorder of the full patch coincide. Between the coinciding regions therealways remain strips, often called “worms,” containing patterns that donot match. If we could see the entire infinite pattern we would observethat any bounded region in the rotated pattern can be found in theunrotated pattern, but the larger the region the further away we have tolook in order to find it. Even so, there is no translation that willbring the whole infinite red pattern into full coincidence with theinfinite blue one.

It is possible to make two patterns that are not identical that containthe same statistical distribution of bounded substructures of arbitrarysize. Bound specific patches of two Penrose patterns that differ by a10-fold rotation both contain roughly the same number of 5-fold starsand roughly the same number of such stars that are surrounded by fivethin rhombic shapes. The two patterns contain the same statisticaldistribution of bounded structures at all scales. Two patterns that arestatistically the same in this sense are called “indistinguishable.” Forpatterns with quaisperiodic order, a symmetry operation is one thattakes the pattern into an indistinguishable one. Quasiperiodic order maybe created in three-dimensional space and the definition is not intendedto be limiting to two-dimensional order.

A “photonic quasicrystal” means a quasicrystal that is capable ofallowing the transmission, steering, manipulation, and control of someelectromagnetic radiation. It is not intended that the term be limitedto the transmission of electromagnetic radiation in the visible region.It is also not intended to be capable of transmitting of allelectromagnetic radiation. In some preferred embodiments, the photonicquasicrystal refracts, reflects, defracts, or absorbs electromagneticradiation at individual frequencies.

A “heterostructure” means a dielectric structure with one or moreinterface(s) across which the chemical composition changes. Theinterface of the two dielectrics contains a scattering centre in whichlight propagates more slowly. If the scattering centers are regularlyarranged in a medium, light is coherently scattered. In this case,interference causes some frequencies not to be allowed to propagate,giving rise to forbidden and allowed bands. Regions of frequency mayappear that are forbidden regardless of the propagation direction.

A “photonic bandgap” material or structure means that for a certainrange of wavelengths, no states exist in the structure forelectromagnetic radiation to occupy.

Electromagnetic radiation with these wavelengths is forbidden in thestructure and cannot propagate. The presence of a single point defect,i.e., part of the structure in which the electromagnetic radiation canpropagate, generally results in a “localized state”, i.e., a tightlyconfined region of light energy which must stay within the defect, sinceit cannot propagate in the structure, and provided the energy is notbeing absorbed by the material. By introducing defects, one canintroduce allowed energy levels in the gap. Defects, appropriatelydesigned and arranged, can create waveguides with directional control(e.g., one micron radius, 90 degree bends with 98 percent transmissionefficiency), drop/add filters, multiplexors/demultiplexors, resonators,and laser cavities.

A “quasicrystalline lattice structure” means that the quasicrystal is inthe form of material patterned with an open framework. For example, inpreferred embodiments, the quasicrystal is made by stereolithography inwhich polymerization produces rhombohedral cells characterized by rodsthat creates an open framework.

A “dielectric resonator” or “cavity” means a device arranged that allowselectromagnetic radiation to propagate back and forth and build upintensity. An “optical resonator” or “resonant optical cavity” means anarrangement of optical components which allow a light beam to propagateback and forth and build up intensity. For example, if a mirror ispartly transparent one can feed light from outside into the cavity. Twohighly reflecting low-less reflectors may be positioned with theirreflecting surfaces facing one another to form the cavity. A collimatedlaser beam enters the cavity and the wavelength of the incident light israpidly swept in time. At specific wavelengths and at specific featurepositions, light resonates within the cavity, building up energy,corresponding to a peak in the transmitted light.

“Visible spectrum” means electromagnetic radiation that ranges fromapproximately 780 nanometers (abbreviated nm) to approximately 380 nm. Aregular incandescent bulb produces light within the visible spectrum. Italso wastes a lot of its energy radiating invisible radiation, too. Thephotonic quasicrystal can be tailored so that it radiates almost all ofits light in the desired visible with little (not really zero) waste.This is the reason for saying “substantially only” with regard toemitting light in the visible region.

An “interconnect” means a physical attachment between two or moreobjects.

A “preselectable rotational symmetry having a characteristic photonicbandgap structure forbidden in a crystalline material” means thestructure has at least one five-, seven- or higher-fold symmetry axisand whose bandgap structure exhibits this same symmetry. For example, aphotonic quasicrystal can have three-dimensional icosahedral symmetry ortwo dimensional seven-fold symmetry, either of which are impossible forphotonic crystals.

As used herein, a material having a spherically symmetric property meansthat rotation by any angle in three dimensions produces no change withregard to the physical property. Circularly symmetric means thatrotation by any angle two dimensions produces no change. Bandgaps arenot precisely spherically symmetric (same for circularly symmetric);thus, usage is intended to be substantially so.

A “quasicrystalline structure includes higher point group symmetry thana crystalline counterpart” means that the quasicrystal can have seven-,eight-, and higher-fold symmetry axes, whereas periodic crystals cannever have greater than six-fold symmetry.

For example, a crystal, planar hexagonal lattice has six-fold symmetry,the highest and most circular symmetry possible for a crystal orperiodic pattern. Quasicrystals allow higher, more circular symmetries,such as patterns with seven-, eleven-, forty-seven- or even highersymmetries, Similarly in three-dimensions, the highest symmetry possiblefor a periodic pattern or crystal is cubic symmetry, whereasquasicrystals can have icosahedral symmetry, which includes five-foldsymmetries and which is more spherically symmetric.

“Achieving higher power efficiency over a selected range of frequencies”means that more of the input power is converted into radiation of thedesired frequencies and less is wasted in producing undesiredfrequencies. For example, an ordinary microwave antenna also broadcastsnear infrared and radio waves which are not useful for microwavetransmission, but a quasicrystalline antenna would stop the unneededwaves and refocus their power towards producing more microwave radiationat the useful frequencies.

A “sub-wavelength quasicrystalline structure” means the spacing betweenthe repeating elements in the quasicrystalline structure is smaller thanthe wavelength of the light.

A “stealth material” means a radar absorbent material that absorbs theincoming radar radiation without producing any reflections.

When light strikes a substance, some absorption and some reflection takeplace. Substances that transmit almost all the light waves that fallupon them are said to be transparent. A transparent substance is onethrough which you can see clearly. Clear glass is transparent because ittransmits light rays without scattering them. Transparent media such asglasses are isotropic, i.e., light behaves the same way no matter whichdirection it is traveling in the medium. Substances through which somelight rays can pass but through which objects cannot be seen clearlybecause the rays are scattered are called translucent. Substances thatdo not transmit any light rays are called opaque.

The angle at which the beam is reflected depends on the angle at whichit strikes the mirror. The beam approaching the mirror is the incidentor striking beam, and the beam leaving the mirror is the reflected beam.Any type of mechanism used to reflect light is called a reflector. Theterm “reflected light” simply refers to light waves that are neithertransmitted nor absorbed, but are thrown back from the surface of themedium they encounter. When light is reflected from a mirror, the angleof reflection of each ray equals the angle of incidence. When light isreflected from a substrate of an uneven surface, the reflected beam isscattered, or diffused. Because the surface of the substrate is notsmooth, the reflected light is broken up into many light beams that arereflected in different directions.

An antenna is a conductor or a set of conductors used either to radiateelectromagnetic radiation into space or to collect electromagneticradiation from space. A transmitter is a device that generateselectromagnetic radiation. A signal travels through a transmission lineto an antenna. The antenna converts the signal into waves that radiateinto space from the antenna. The waves travel through space. If anotherantenna is placed in the path of the waves, it absorbs part of the wavesand converts them to a signal that travels through another transmissionline and is fed to a receiver.

The term waveguide means any type of transmission line in the sense thatit is used to guide electromagnetic radiation from one point to another.Typically, the transmission of electromagnetic energy along a waveguidetravels at a velocity somewhat slower than electromagnetic radiationtraveling through free space. A waveguide may be classified according toits cross section (rectangular, elliptical, or circular), or accordingto the material used in its construction (metallic or dielectric). Glassfibers, gas-filled pipes, and tubes with focusing lenses are examples ofoptical waveguides.

As used herein, the dielectric constant means the extent to which asubstance concentrates the electrostatic lines of flux. Substances witha low dielectric constant include a perfect vacuum, dry air, and mostpure, dry gases such as helium and nitrogen. Materials with moderatedielectric constants include ceramics, distilled water, paper, mica,polyethylene, and glass. Metal oxides, in general, have high dielectricconstants.

The refractive index is related to the dielectric constant of amaterial. The refractive index (or index of refraction) of a material isthe factor by which the phase velocity of electromagnetic radiation isslowed in that material, relative to its velocity in a vacuum. Althoughit is not intended that the invention be limited by any particularmechanism, it is believed that an electromagnetic wave's phase velocityis slowed in a material because the electric field creates a disturbancein the charges of each atom (primarily the electrons) proportional tothe permittivity. The charges, in general, oscillate slightly out ofphase with respect to the driving electric field. The charges thusradiate their own electromagnetic wave that is at the same frequency butwith a phase delay. The macroscopic sum of all such contributions in thematerial is a wave with the same frequency but shorter wavelength thanthe original, leading to a slowing of the wave's phase velocity. Most ofthe radiation from oscillating material charges will modify the incomingwave, changing its velocity.

If the refractive index of a medium is not constant, but variesgradually with position, the material is known as a gradient-indexmedium. The refractive index of certain media may be different dependingon the polarization and direction of propagation of the electromagneticradiation through the medium. A strong electric field of high intensityelectromagnetic radiation (such as the output of a laser) may cause amedium's refractive index to vary as the electromagnetic radiationpasses through it, giving rise to nonlinear optics. Thus, in preferredembodiments, the invention relates to a material that contains regionswith changing refractive indices that are arranged in quasicrystallineorder.

A material having a “plurality of refractive indices” means that thematerial does not have a single uniform refractive index whentransmitting electromagnetic radiation along a single axis. As usedherein, a “plurality” means two or more. This may be the result of thematerial being constructed from two or more materials or because themolecular components of the material are arranged in varyingorientations or constitution along the axis. In certain preferredembodiments, the refractive index of the material is continuouslyvarying in waves from a maximum value, i.e., because the electromagneticradiation is slowed, to a minimum value, i.e., the electromagneticradiation is traveling at its fastest velocity, through the material ina pattern that is quasiperiodic. A plurality of medium values means thatthe refractive indices are a value in between the maximum and minimumvalues, preferably an average or median value. It is not intended thatthe maximum, minimum, and medium values be entirely consistent, but itis sufficient that they are similar in relative values to the extentthat the production methods employed result in slight deviations.

Optical interconnections and optical integrated circuits have severaladvantageous over their electronic counterparts. They are free fromelectrical short circuits. They have low-loss transmission and providelarge bandwidth; i.e., multiplexing capability, capable of communicatingseveral channels in parallel without interference. They are capable ofpropagating signals within the same or adjacent fibers with essentiallyno interference or cross-talk. They are compact, lightweight, andinexpensive to manufacture, and more facile with stored information thanmagnetic materials. Hybrid electro-optical components are limited solelyby the speed of their electronic parts.

Photodiodes are devices that utilize the photoelectric effect, i.e.,emission of electrons from matter upon the absorption of electromagneticradiation, including p-n junction or p-i-n structures. When light withsufficient photon energy strikes a semiconductor, photons can beabsorbed, resulting in generation of a mobile electron and electronhole. If the absorption occurs in the junction's depletion region, thesecarriers are swept from the junction by the built-in field of thedepletion region, producing a photocurrent. Photodiodes can be used ineither zero bias or reverse bias. In zero bias, light falling on thediode causes a voltage to develop across the device, leading to acurrent in the forward bias direction. This is called the photovoltaiceffect, and is the basis for solar cells.

Diodes usually have relatively high resistance when reverse biased. Thisresistance is reduced when light of an appropriate frequency shines onthe junction. Hence, a reverse biased diode can be used as a detector bymonitoring the current running through it. Circuits based on this effectare more sensitive to light than ones based on the photovoltaic effect.P-N photodiodes are used in similar applications to otherphotodetectors, such as photoconductors, charge-coupled devices, andphotomultiplier tubes.

Photonic quasicrystals act as optical devices that make it possible tocontrol which bands of photons pass through the structure and which onesare blocked. An icosahedral photonic quasicrystal may be aquasicrystalline array of two or more materials with differentdielectric constants (or, different indices of refraction) that hasnearly spherically symmetrical photonic bandgaps. Icosahedral photonicquasicrystals are preferred for three-dimensional (3D) applications.Quasicrystals with high order rotational symmetry are preferred for manytwo dimensional (2D) applications.

Photonic crystals have a variety of technological uses due to theirstopgaps (also called bandgaps) that block light of certain frequenciesfrom passing along certain directions. Although such crystallinestructures allow many technological applications to be fulfilled, theirperiodic translational symmetry constrains the stopgap structure to haveonly certain rotational symmetries and frequencies. Quasicrystallineheterostructures can have arbitrary rotational symmetries in two andthree dimensions. Many of the most important applications of photonicmaterials require blocking the passage of light in all directions,effectively trapping the light inside the material. Becausequasicrystals may have higher rotational symmetries, their stopgapstructure can be more spherically symmetric than that of any photoniccrystal, facilitating the formation of stopgaps in all directions andenabling practical devices to be constructed from a wider variety ofmaterials. Furthermore, because they have quasiperiodic (rather thanperiodic) translational symmetry, they can have more diverse andcontrollable stopgap structures.

The quasicrystals may be constructed from two or more types ofdielectric material arranged in a quasiperiodic pattern whose rotationalsymmetry is forbidden for periodic crystals (such as five-fold symmetryin the plane and icosahedral symmetry in three-dimensions). Becausequasicrystals often have higher point group symmetry than ordinarycrystals, their gap center frequencies are closer and the gaps widthsare more uniform, optimal conditions for forming a complete bandgap thatis more nearly spherically symmetric. Although previous studies havefocused on 1D and 2D quasicrystals, where exact (1D) or approximate (2D)bandstructures can be calculated numerically, analogous calculations forthe 3D case are computationally challenging and have not been performedto date. To circumvent the computational problem, stereolithography hasbeen used to construct a photonic quasicrystal with centimeter-scalecells and to perform microwave transmission measurements. The 3Dicosahedral quasicrystals exhibit sizeable stopgaps and, despite theirquasiperiodicity, rather uncomplicated spectra which allow experimentaldetermination of the faces of their effective Brillouin zones. Theseresults suggest that 3D icosahedral quasicrystals can act as photonicbandgap materials. In preferred embodiments, the characteristic lengthof a photonic band-gap quasicrystal matchs the range of wavelengths ofelectro-magnetic radiation it interacts with, e.g. cm for microwaves,1-10 μm for near infrared, and 0.4-0.8 μm for visible light.

In 1984, Schechtman et a.l Metallic phase with long-range orientationalorder and no translational symmetry, Phys. Rev. Lett., 53, 1951-1953(1984) observed icosahedral symmetry with five-fold rotation axes in theelectron diffraction pattern of an alloy of Al—Mn. Simultaneously, theconcept of long range quasiperiodic order with icosahedral symmetry wastheoretically developed by Levine and Steinhardt, Quasicrystals: A NewClass of Ordered Structures, Phys. Rev. Lett. 53, 2477-2480 (1984).

A photonic icosahedral quasicrystal is shown in Fig. la. The diamondstructure shown in FIG. 1b was made for comparative experiments; diamondhas been suggested as an optimal structure for photonic crystals.Photonic crystals are based on the fact that photons Bragg scatter froma medium with a periodically modulated refractive index. Multiplescattering at frequencies near the Bragg condition prevents propagationin these directions, producing a “stopgap.” Overlap of the stop gaps inall directions yields a complete photonic bandgap and traps the light. Acomplete overlap occurs more readily in more isotropic structures.

Quasicrystals have long-range quasiperiodic order and higher point groupsymmetries, so photons Bragg scatter along a more spherically symmetricset of directions. As the symmetry increases, the “Brillouin zone”becomes more circular or more spherical. Photonic quasicrystals alsoallow for a higher degree of flexibility and tuneability for defect modeproperties.

FIG. 1c shows the effective Brillouin zone (related to the Pseudo-Joneszone considered in electronic transport in quasicrystals) of theicosahedral structure with its irreducible Brillouin zone highlighted inyellow. For comparison, FIG. 1d shows the first Brillouin zone of thediamond (FCC) structure with its irreducible Brillouin zone. As ameasure of sphericity, along the edge of diamond structure's irreducibleBrillouin zone, the magnitude of k (which is proportional to stop gapcenter frequency in first order approximation) increases 29.1% from L toW. Along the edge of the effective irreducible triacontahedral Brillouinzone of the icosahedral structure, the magnitude of k increases only17.5% from the two fold to the five fold symmetry points. Moreover thetriacontahedron's faces are identical and subtend smaller solid angles.

The photonic bands of our rod decorated diamond lattice were obtainedusing the MIT Photonic-Bands package. FIG. 3a shows the calculated bandstructure along a rotation about a 2-fold axis. The rotation path isillustrated by the dotted red line in FIG. 1d . There is excellentagreement between the observed and calculated gap positions.

From the rather complex spectra, the gaps appear to result from Braggscattering. A wavevector, which resides on the plane defined by areciprocal lattice vector G is Bragg scattered by G. Such a wavevectorsatisfies the condition k·G=|G|²/2 or equivalently, |k|=|G|/(2cos(θ)).To lowest order, the center frequency of a stopgap is thereforef_(G)=(c/n)|G|/(4πcos(θ)), where c is the speed of light and n is theBruggeman effective medium index [14]. The dashed curves in FIGS. 2a and3b correspond to a) 1/cos(θ−θ_(o)) angular dependence, hence Braggscattering.

Compared with the diamond structure, the quasicrystal spectrum appearsless complex. Since the scattering function for a quasicrystal is adense set of Bragg spots (of zero measure), many gaps and zone facesintersecting might be expected. On the contrary, there appear to be afew well defined 1/cos(θ−θ_(o)) curves in FIG. 2A-B and therefore fewzone boundaries with sizable gap formation. As a novel method forvisualizing the effective Brillouin zone structure, the process wasinverted by using the gaps to find the zone faces. The points inreciprocal space responsible for the gaps are located by assuming|k)θ)|˜nf(θ)/(2πc) and polar plots of T(r=f, θ=θ) are made (FIG. 4A-E).For the diamond lattice, data from a rotation about a 4-fold axis(dashed line in FIG. 1d ) and a 2-fold axis (dotted line in FIG. 1d )are shown in FIG. 4b and FIG. 4c , respectively. FIG. 4a . shows thecalculated frequency deviation (|δf|=|(f−(c/n)|k∥/2π)) vs. wavevector ofthe 4-fold rotation.

Transmission data for the quasicrystal is shown in FIGS. 4d and 4e . Thefact that the low transmission regions correspond to straight linesindicates that the gaps lie on planes. These transmission polar plots,without any further analysis, directly provide the scattering planes andthe effective Brillouin zones. In the smallest zone in FIGS. 4d and 4e ,the decagon is seen from the 5-fold rotation, and the additionalsymmetry planes are seen from the orthogonal 2-fold rotation, whichcorrespond to the respective cuts of the triacontahedral Brillouin zoneshown in FIG. 1 c. (The wavevector corresponding to the edge center ofthe smallest visible decagon in FIG. 4d is τ²/√{square root over (τ²+1)}in units of 2π/d, where d is the rod length.) There are however, severalunexpected features: a strong scattering plane along a 45° direction,the absence of strong scattering from the “2⊥” plane (dashed line; whichmay be a polarization effect due to the rod decoration of the unitcells), and another strong scattering plane (dash-dotted line) not onthe triacontahedron. A complete photonic bandgap would result if thedotted blue curves, (cuts of a constant frequency sphere) were containedwithin the gap of the zone boundary.

These results establish the following. First, there is a relativelywell-defined effective Brillouin zone, all of whose faces are consistentwith the quasicrystal Bragg pattern. Second, the Brillouin zonestructure is surprisingly simple despite the fact that a quasicrystalhas a dense set of Bragg spots. Third, the measured Brillouin zone isclose to spherical, with the largest difference in gap centercorresponding to 17% (dotted curve in FIG. 2a ). Also, the experimentsdemonstrate that three-dimensional quasicrystals exhibit sizablestopgaps on reasonably well-defined effective Brillouin zone faces. Forpreferred embodiments a smoother, more spherical multiply connected unitcell decoration with more equal filled/void ratio would reducepolarization effects and enhance the gap overlap while maintaining thenearly spherical Brillouin zone. Laser tweezers used for particletrapping or two-photon polymerization would allow construction of aquasicrystalline matrix of dielectric components with a photonic bandgapin the visible.

Photonic quasicrystals can be used for many purposes and to produce avariety of compositions and products including but not limited to lamps,compact optical circuits, optical filters, efficient antenna substrates,mirrors, stealth materials, dielectric resonators, semi-conductors,patterned materials. Such compositions and products can be 2D or 3Dobjects and have photonic bandgap materials composed of quasicrystals.Any material that can be used in the production of photonic crystals canbe used to make photonic quasicrystals. Products, compositions andmaterials produced from photonic quasicrystals are an improvement overthose made from photonic crystals, at least in part because of thebandgap properties of the quasicrystalline structures.

Methods to produce photonic crystals can be adapted to produce photonicquasicrystals by determining the desired quasiperiodic structure that isdesired for a particular application. For example, liquid polymers canbe photopolymerized at discrete points using computer-directed lasers tocause two-photon polymerization and thus produce discrete structuresbathed in a liquid of unpolymerized monomers (that are then removed orwashed) away. In accordance with this invention, 3D structures usingicosahedral quasicrystals exhibit near spherical symmetry and areparticularly useful.

By way of example, incandescent lamps are highly inefficient and produceelectromagnetic radiation over a broad frequency range, most of which isunseen by the human eye. A lamp constructed from a photonic quasicrystalcan be designed to only produce radiation within the visible spectrum,resulting in a highly efficient conversion of input power into usefulvisible luminosity. Preferably such lamps are made from a 3D photonicquasicrystals, and more preferably from icosahedral photonicquasicrystals.

A photonic quasicrystal can be used as a reflector to enable a lightbeam in a photonic circuit to be reflected at angles greater than thecritical angle and thus create compact optical circuits. Such photonicbandgap devices can be a two- or three-dimensional quasicrystallinearray of dielectric elements, or, for the two-dimensional case, an arrayof columnar holes formed in a substrate. The holes are filled with airor another material having a different dielectric constant than thesubstrate. The quasicrystal structure allows angles of reflection andcombinations of frequency bandgaps that are not possible for crystals.For example, the compact optical waveguide of U.S. Pat. No. 6,134,369can be adapted to use photonic quasicrystals.

An optical filter includes a dielectric layer formed within a resonantoptical cavity, with the dielectric layer having formed therein asub-wavelength periodic structure to define, at least in part, awavelength for transmission of light through the resonant opticalcavity. The optical filter has applications to spectrometry,colorimetry, and chemical sensing. For example, optical filters of theinvention can be made from photonic quasicrystals using methodsdescribed for photonic crystal analogues as set for the in U.S. Pat. No.5,726,805.

A broadband antenna system utilizes multiple photonic bandgap crystalsto achieve higher power efficiency over a selected range of frequencies.Multiple custom tailored photonic bandgap quasicrystals can be used toform a substrate for the antenna system. Each of the quasicrystals isdesigned to cover a specific range of frequencies. The multiplequasicrystals are attached together to form a photonic bandgap substratewhose bandwidth varies as a function of location on the substrate.Photonic quasicrystals enable a range of frequency bands and angles ofreflection that are impossible for crystals. For example, antennasystems of the invention can be made from photonic quasicrystals usingmethods described for photon crystal analogues as set for the in U.S.Pat. No. 5,541,613.

An optical circuit and waveguide can be fabricated on a substrate havinga two-dimensional quasiperiodic dielectric structure with arbitrarysymmetry. The quasiperiodic dielectric structure exhibits a range offrequencies of electromagnetic radiation which cannot propagate into thestructure and are not possible for photonic crystals. Radiation at afrequency within the frequency bandgap of the structure is confinedwithin the circuit and waveguide by the quasicrystalline dielectricstructure surrounding the circuit and waveguide. Radiation losses aresubstantially eliminated. For example, optical circuit and waveguides ofthe invention can be made from photonic quasicrystals using methodsdescribed for photon crystal analogues as set for the in U.S. Pat. No.5,526,449.

A photonic bandgap mirror having a quasicrystalline lattice structure isplaced in front of light emitted from one side of the laser diode array.If the photonic quasicrystal has an incomplete bandgap, light parallelto the symmetry axis will be primarily reflected while light at selectangles—depending on the symmetry of the quasicrystal—is transmittedthrough the mirror. Such a mirror is useful for making a laser cavity.The photonic quasicrystal allows reflection and transmission angles thatare impossible for photonic crystals. For example, quasicrystal mirrorsof the invention can be made from photonic quasicrystals using methodsdescribed for photon crystal analogues as set for the in U.S. Pat. No.5,365,541.

Photonic quasicrystals can be designed as material that trapselectromagnetic radiation at programmable ranges of frequencies nearlyuniformly at all angles and thus has use for stealth applications.

A dielectric resonator comprised of a resonant defect structure placedin a quasicrystal lattice of dielectric elements confineselectromagnetic energy within a frequency band in the photonic bandgap.The frequency band of the confined electromagnetic energy is thuscontrollable based on the selected quasicrystal lattice. For example,dielectric resonators of the invention can be made from photonicquasicrystals using methods described for photon crystal analogues asset for the in U.S. Pat. No. 5,187,461.

EXAMPLES Example 1. Stereolithography Created Lattices

A D-dimensional periodic lattice has D integer independent basisvectors, while D+N integer linearly independent vectors (with integerN≥1) are required to describe the quasicrystal lattice. The icosahedralquasicrystal lattice of points can be constructed by projecting thepoints of a six-dimensional hypercubic lattice, the six-dimensionalanalog of a three-dimensional cubic lattice. The coordinate of anylattice point can be described by the relation: R=Σ_(i=1) ⁶n_(i)a_(i),where the n_(i) are a subset of the integers and a_(i) are the 6 basisvectors: a₁=(1,τ,0), a₂=(−1,τ,0), a₃=(0,1,τ), a₄=(0,−1,τ), a₅=(τ,0,1),a₆=(τ,0,−1), and τ=(√5−1)/2 the golden mean. The structure has twelve5-fold, fifteen 3-fold and thirty 2-fold symmetry axes. The latticepoints of the icosahedral structure are generated and a solid structurecreated by using equal length rods to connect points in pairs. Theoverall shape shown in FIG. 1a is a dodecahedron, so that each of the 12outside faces is perpendicular to a 5-fold rotation axis.

Crystals are created with a stereolithography machine (model SLA-250from 3D Systems®) that produces a solid plastic model by UV laserphotopolymerization. The resolution is 0.1 mm lateral and 0.15 mmvertical. Both the icosahedral quasicrystal and diamond crystal havevertices connected with rods, of length d=√{square root over (3)}a/4=1cm. The rod diameter is 0.15 cm for the quasicrystal and 0.4 cm for thediamond structure. The quasicrystal has 694 cells, 4000 rods, andconsists of 17.3% volume percent polymer whereas the diamond structurehas 500 cells and is 7.36% polymer. The refractive index of thepolymerized SLA5170 resin was measured by placing a solid block in awaveguide and recording the transmission and reflection spectrum. For 33GHz microwaves (λ=0.91 cm in air), ñ=1.65−0.025i. The resulting (1/e)absorption length is 12 wavelengths. (The finite absorption from thepolymer reduces the transmission approximately as exp(−2ωηL/c), whereη□=0.025 is the imaginary part of the refractive index and L is thetransmission path length. The actual attenuation will depend on thegeometry and the modes. All curves in FIG. 2A-B have been multiplied bythe same simple exponential factor to reduce the background slope. Thishas no effect on the gap determination.)

Example 2. Transmission Measurements

Transmission measurements for the icosahedral quasicrystal of FIG. 1acan be made with a HP Model 8510C Vector Analyzer in three bands, from 8to 15, from 15 to 26 and from 26 to 42 GHz. To approximate plane waves,a single TE₁₀ mode is coupled through two sets of horn-attachedwaveguides with two custom-made polystyrene microwave lenses asschematically shown in the insert in FIG. 2A-B. Before the sample is putin, the transmission spectrum of the setup is recorded fornormalization. Since the sample has different symmetry and dimensions indifferent directions, the transmission spectrum should be orientationand polarization sensitive. The sample is aligned so that the incidentbeam is perpendicular to one of the sample's rotational symmetry axes.The sample is rotated along that rotational symmetry axis, and therelative transmission spectrum is recorded, for example, every 2degrees. For the quasicrystal, a rotation about the 2-fold axis coversall the external points of the irreducible Brillouin zone and bothpolarizations. The region covered by the rotation about the 2-fold(5-fold) axis is shown as the dotted (dashed) red line in FIG. 1 c. InFIG. 2A-B, the measured transmission T(f,θ) for this rotation are shownin overlapping plots from two frequency bands.

Example 3. Holographic Optical Trapping

The foregoing photonic quasicrystal is macroscopic, however, this designcan be miniaturized,. e.g., for nanoparticles or other particles. Forexample, laser tweezers can be used for particle trapping or two-photonpolymerization can be used. Such methods allow construction of aquasicrystalline matrix of dielectric components with a photonic bandgapin the visible.

Computer generated holograms are projected through ahigh-numerical-aperture microscope objective lens to create largethree-dimensional arrays of optical traps. Light at 532 nm form afrequency-doubled diode-pumped solid state laser (Coheret Verdi) isimprinted with phase-only holograms using a liquid crystal spatial lightmodulator (SLM) (Hamamatsu X8267 PPM). The modified laser beam isrelayed to the input pupil of a 100×NA 1.4 SPlan Apo oil immersionobjective mounted in an inverted optical microscope (Nikon TE-2000U),which focuses it into traps. The same objective lens is used to formimages of trapped objects using the microscope's conventional imagingtrain.

Colloidal silica microspheres (1.53 micrometers in diameter, DukeScientific Lot 5238) dispersed in an aqueous solution of 180:12:1(wt/wt) acrylamide, N, N′-methylenebisacrylamide, anddiethoxyacetophenone (all Aldrich electrophoresis grade). This solutionphotopolymerizes into a transparent polyacrylamide hydrogel underultraviolet illumination. Fluid dispersions were imbibed into 30micrometer thick slit pores formed by bonding the edges of number 1coverslips to the faces of glass microscope slides. The sealed sampleswere then mounted on the microscope's stage for processing and analysis.

Silica spheres are roughly twice as dense as water and sediment rapidlyinto a monolayer above the coverslip. A dilute layer of spheres isreadily organized by holographic optical tweezers into two andthree-dimensional quasicrystalline configurations. Holographic opticaltraps were used to create nonuniform architectures with specificallyengineered features, e.g., a channel with a ninety-degree turn embeddedin a quasiperidic pattern having 8-fold symmetry. The quasicrystals aregelled.

Example 4. Fabrication of Quasicrystals with Arbitrary Symmetry andGeometric Complexity

In order to create quasicrystal photonic structures with arbitrarycomplexity and symmetry one uses a system to write a pattern directly inthree dimensions. One uses a dye sensitized epoxy resist which willpolymerize when exposed to sufficient light intensity. One uses acomputer program to control a laser and associated optics to createspatial modulations of light intensity to write a quasicrystallinepattern. One directs the light to certain positions by focusing andscanning as with laser tweezers, or one broadly addresses a volume ofsaid sensitized photopolymer by interfering two or more laser beams tocreate a modulated intensity pattern. One directly writes thequasicrystalline pattern with all its symmetry or one makes an exposureand then repeats the exposure after rotation of the light beam or thesample to provide the five fold, seven fold or higher fold symmetryrequired. One controls the polymerization in a two-step reaction(exposure and postexposure bake). The two-step process allows one toexpose the entire material without distorting or perturbing the lightpaths. One then bakes the exposed material to allow solidification ofthe exposed regions. One completes the process by drainage of theunderexposed liquid. This leaves a dielectric constant contrast betweenthe photopolymer and the vacuum/air. One obtains different contrasts, ormore gradual continuous variations in dielectric constant by backfilling the void space with material of a different dielectric constant.One chooses a liquid material which diffuses into the photopolymer tocreate the more gradual or continuous dielectric variation. One adjuststhe intensity of the light exposure to control the ability of the solidphotopolymer to absorb the liquid and to swell to accommodate the liquidand locally provide an average dielectric constant between that of thephotopolymer and the liquid. The controllable continuous dielectricconstant results from the local exposure of the photopolymer and theamount of time the liquid is allowed to diffuse before it is solidifiedby heat or light treatment.

The polymerization is controlled in a two-step reaction (exposure andpostexposure bake) without perturbing the interference patterns. Wecreated a series of 2D and 3D defect-free porous structures with periodsof 0.9-8˜\micron in an area larger than 1˜\unit{mm} by usingvisible-laser-initiated cationic polymerization of epoxides. Epon SU-8 san epoxide abundant monomer with a2,2′-((1-methylethylidene)bis(4,1-phenyleneoxymethylene))bis-oxiranestructure.

The initiating system typically includes a photosensitizer, whichabsorbs the visible light and electron transfers to an onium salt viathe formation of a charge-transfer complex to generate the acids. Thephotoacids initiate ring-opening reactions of epoxy groups and the acidsare regenerated in the subsequent steps. The polymerization is thuschemically amplified, resulting in a highly cross-linked film. Twopreferred photosensitizers include the xanthene dyes,2,4,5,7-tetraiodo-6-hydroxy-3-fluorone and Rose Bengal.

The relatively high glass transition temperature of SU-8, 50° C., isdesirable for minimizing the acid diffusion before the postexposurebake. Polymerization during exposure is not desired because it disturbsthe original interference pattern because of the change of refractiveindex of the cross-linked film. When the film is exposed at roomtemperature and then baked, the beam interference stage is separatedfrom the film cross-linking stage. Triethylamine (TEA) is in the resist.At the molar ratio of TEA/2,4,5,7-tetraiodo-6-hydroxy-3-fluorone of0.03:1, respectively open holes were observed, indicating theneutralization of background acids and the resulting partial removal ofthe crosslinked background.

In a further example, one uses laser ablation for the synthesis ofquasicrystals. One deposits films comprised of a preselected materialonto a solid support and ablate via repetitive pulsing with a lasersource such as a pulsed excimer laser. Riabinina et al. (2006)“Photoluminescent silicon nanocrystals by reactive laser ablation”Applied Physics Letters 88, Art. No. 073105.

Still another example one uses lithography to fabricate quasicrystalswith the physical properties desired. One uses photomask lithography tointroduce quasicrystals with the capability of phase-shiftingelectromagnetic radiation. Chang et al. (2006) “Fabrication of photonicbandgap structures with designed defects by edge diffractionlithography” Nanotechnology 17, 133-1338.

It will be appreciated by those skilled in the art that variousomissions, additions and modifications may be made to the inventiondescribed above without departing from the scope of the invention, andall such modifications and changes are intended to fall within the scopeof the invention, as defined by the appended claims. All references,patents, patent applications or other documents cited are hereinincorporated by reference.

1-4. (canceled)
 5. A method comprising: a) providing a quasicrystalline composition comprising material elements with physical properties in which waves are oriented in different directions; and b) arranging said material elements into a quasi-periodic pattern that modulates sound waves.
 6. A device comprising a quasicrystalline composition comprising material elements with physical properties in which waves are oriented in different directions, wherein said material elements are arranged in a quasi-periodic pattern that modulates sound waves. 